218 research outputs found
Virtual Knot Theory --Unsolved Problems
This paper is an introduction to the theory of virtual knots and links and it
gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen
Infinitely many two-variable generalisations of the Alexander-Conway polynomial
We show that the Alexander-Conway polynomial Delta is obtainable via a
particular one-variable reduction of each two-variable Links-Gould invariant
LG^{m,1}, where m is a positive integer. Thus there exist infinitely many
two-variable generalisations of Delta. This result is not obvious since in the
reduction, the representation of the braid group generator used to define
LG^{m,1} does not satisfy a second-order characteristic identity unless m=1. To
demonstrate that the one-variable reduction of LG^{m,1} satisfies the defining
skein relation of Delta, we evaluate the kernel of a quantum trace.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-18.abs.htm
On the Links-Gould invariant and the square of the Alexander polynomial
This paper gives a connection between well chosen reductions of the
Links-Gould invariants of oriented links and powers of the Alexander-Conway
polynomial. We prove these formulas by showing the representations of the braid
groups we derive the specialized Links-Gould polynomials from can be seen as
exterior powers of copies of Burau representations.Comment: 19 page
A cubic defining algebra for the Links-Gould polynomial
We define a finite-dimensional cubic quotient of the group algebra of the
braid group, endowed with a (essentially unique) Markov trace which affords the
Links-Grould invariant of knots and links. We investigate several of its
properties, and state several conjectures about its structure
On a Relation Between ADO and Links-Gould Invariants
In this thesis we consider two knot invariants: Akutsu-Deguchi-Ohtsuki(ADO) invariant and Links-Gould invariant. They both are based on Reshetikhin-Turaev construction and as such share a lot of similarities. Moreover, they are both related to the Alexander polynomial and may be considered generalizations of it. By experimentation we found that for many knots, the third order ADO invariant is a specialization of the Links-Gould invariant. The main result of the thesis is a proof of this relation for a large class of knots, specifically closures of braids with five strands
On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants
We study various specializations of the colored HOMFLY-PT polynomial. These
specializations are used to show that the multivariable link invariants arising
from a complex family of sl(m|n) super-modules previously defined by the
authors contains both the multivariable Alexander polynomial and Kashaev's
invariants. We conjecture these multivariable link invariants also specialize
to the generalized multivariable Alexander invariants defined by Y. Akutsu, T.
Deguchi, and T. Ohtsuki.Comment: 16 page
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